a) Field of the Invention
The present invention relates generally to the field of analog encoding and filtering and more particularly to solving systems of differential equations by encoder and filter circuits using time encoding and pulse domain signals.
a) Description of Related Art
Conventionally, differential equations could be solved in the (1) original analog domain via analog amplifiers or in the (2) digital domain after an Analog to Digital Converter (ADC) digitization. In the analog domain, a disadvantage of the conventional approach is that accuracy is severely limited by dynamic range of the analog components such as the feedback analog amplifiers. Conventional circuits for solving differential equations in the analog domain are based on analog computation where the information is encoded in the amplitude of voltages or current signals. Accuracy is limited by the analog components, such as operational amplifiers. In the digital domain, the disadvantage of the conventional approach is that speed is limited by the performance of ADC conversion.
Time encoding, that encodes amplitude information into a time sequence, is an alternative to classical sampling and is used for the design of analog to digital converters. A time encoding machine may be implemented as a real-time asynchronous circuit. Time encoding is invertible such that the amplitude information can be recovered from the time sequence with arbitrary accuracy. The amplitude information in a bandlimited signal can be perfectly recovered if the difference between any two consecutive values of the time sequence is bounded by the inverse of the Nyquist rate.
A Hadamard circuit may be used for performing Hadamard arithmetic operations of analog inputs with the solution encoded in the time domain. However the Hadamard circuit may not be used to solve differential equations. Furthermore, the arithmetic operations of the Hadamard circuit can only be performed on analog inputs.
In short, the conventional circuits solve differential equations in the analog domain via the use of analog components. These circuits are limited by the accuracy of these analog components. They are therefore impractical for applications requiring accuracy. Further, they are limited to pulse encoding and do not attempt to do any processing such as solving differential equations using the encoder machine presented. No prior circuit has been proposed to solve a system of linear first order differential equations in the pulse domain.